Autoparts Inc. produces three types of parts' assemblies: (P,Q,R)and they each require (2,5,3) units of work on machine I and (2,2,4) units of work on machine II respectively. The two machines have respectively (160,190) units of work available per day. It is given that their profits are: (10,12,9) dollars respectively.
Assume that the production is given by x,y,z respectively and construct the following:
The profit function to be maximized is:?
The restriction imposed by machine I is:
? <= ?.
The restriction imposed by machine II is: ? <= ?.
Assume that the total production of the product P is required to be greater than or equal to the total production of the other two. Then we get the constraint: x >= ?.
To construct the profit function to be maximized, we need to calculate the total profit based on the production quantities of parts P, Q, and R.
The profit for part P is given as $10 per unit, the profit for part Q is $12 per unit, and the profit for part R is $9 per unit.
So, the profit function can be expressed as:
Profit = 10x + 12y + 9z
To find the restriction imposed by machine I, we need to consider the work units required by each part and the maximum work units available on machine I (160 units).
The work units required by part P on machine I is 2 units.
The work units required by part Q on machine I is 5 units.
The work units required by part R on machine I is 3 units.
To express the restriction imposed by machine I mathematically, we can write:
2x + 5y + 3z <= 160
Similarly, to find the restriction imposed by machine II, we consider the work units required by each part and the maximum work units available on machine II (190 units).
The work units required by part P on machine II is 2 units.
The work units required by part Q on machine II is 2 units.
The work units required by part R on machine II is 4 units.
So, the restriction imposed by machine II can be written as:
2x + 2y + 4z <= 190
Now, to satisfy the constraint that the total production of part P should be greater than or equal to the total production of the other two parts, we have the constraint:
x >= y + z
These equations represent the profit function and the restrictions imposed by machines I and II.